A thought that we all entertain at some time or other is that the course of our lives might have been very different from the way they in fact have been, with the consequence that we might have been rather different sorts of persons than we actually are. A less common, but prima facie intelligible thought is that we might never have existed at all, though someone rather like us did. Arguably, any plausible theory of personal identity should be able to accommodate both possibilities. Certain currently popular Reductionist theories of personal identity, however, seem to be deficient in precisely this respect. This paper explores some Reductionist responses to that challenge.